(It is not obvious that \(L_g\)  is continuous, or even well-defined; see [1]}.) We denote the set of Carathéodory functions depending on space and time by \({H}^k_0([0,T),D;U) L^1([0,T)\times D^k;C_0(U^k))\)  and its dual space by \({H}^{k*}_0([0,T),D;U) L^\infty _w([0,T)\times D^k;\mathcal {M}(U^k))\)  (see e.g. [2]}).
